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Adaptive fuzzy control for a class of unknown fractional-order neural networks subject to input nonlinearities and dead-zones
Accepted Manuscript
Adaptive Fuzzy Control for a Class of Unknown Fractional-Order Neural Networks Subject to Input Nonlinearities and Dead-Zones Heng Liu, Shenggang Li, Hongxing Wang, Yeguo Sun PII: DOI: Reference:
S0020-0255(18)30326-8 10.1016/j.ins.2018.04.069 INS 13612
To appear in:
Information Sciences
Received date: Revised date: Accepted date:
6 March 2017 13 April 2018 21 April 2018
Please cite this article as: Heng Liu, Shenggang Li, Hongxing Wang, Yeguo Sun, Adaptive Fuzzy Control for a Class of Unknown Fractional-Order Neural Networks Subject to Input Nonlinearities and Dead-Zones, Information Sciences (2018), doi: 10.1016/j.ins.2018.04.069
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Adaptive Fuzzy Control for a Class of Unknown Fractional-Order Neural Networks Subject to Input Nonlinearities and Dead-Zones
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Heng Liua,b , Shenggang Lia,∗, Hongxing Wangb , Yeguo Sunb a College
of Mathematics and Information Science, Shaanxi Normal Universtiy, Xi’an 710119, China b Department of Mathematics and Computational Science, Huainan Normal University, Huainan 232038, China
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Abstract
This paper presents an adaptive fuzzy control (AFC) for uncertain fractionalorder neural networks (FONNs) with input nonlinearities and unmodeled dynamics. System uncertainties and unknown parts of the nonlinear input are approximated by fuzzy logic systems (FLSs). Based on some proposed stability analysis criteria for fractional-order systems (FOSs), an AFC is designed to guar-
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antee the asymptotic stability of the controlled system. Fractional-order adaptive laws (FOALs) are constructed to update adjustable parameters of FLSs.
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Our method can be used to control FONNs with/without sector nonlinearities in control inputs. It also allows us to generalize many existing control methods that are valid for integer-order neural networks to FONNs by using the proposed
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method. Finally, the effectiveness of the proposed method is demonstrated by simulation results.
Keywords: Adaptive fuzzy control, sector nonlinearity, fractional-order neural
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network, dead-zone
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1. Introduction Up to now, neural networks (NNs) have received increasing attention in
many fields [2, 13, 17, 28, 29, 34, 35, 45]. The NN that will be considered in this paper is called fractional-order neural network (FONN), which is an extension of ∗ Corresponding author, College of Mathematics and Information Science, Shaanxi Normal Universtiy, Xi’an 710119, China; Email: [email protected]
Preprint submitted to Elsevier
April 23, 2018
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ordinary NNs and have also been well studied [10, 28, 53] because the fractionalorder derivative provides a powerful technique for describing hereditary and memory properties of the system. FONNs have two advantages: one is that the fractional calculus has infinite memory, the other is that it has one more on the control and synchronization of FONNs.
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freedom [3, 27]. Please see [10, 14] and some references therein for recent results Fuzzy logic systems (FLSs) with linguistic information have been effectively
utilized in the control of unknown nonlinear systems [4, 6, 7, 9, 11, 18, 20, 21, 24,
25, 26, 30, 33, 36, 42, 47, 48, 49, 50]. However, there are few works that studied the adaptive fuzzy control (AFC) of FONNs because it is a challenging work to
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check the stability by using a fractional-order stability criterion. On the other
hand, we notice that inputs (particularly, nonlinear inputs), outputs, and state constraints, which are reflected by physical stoppages, dead-zones, backlash, saturation, and safety specifications, are very common in most physical systems, and nonlinear inputs usually lead to instability of the controlled system [7, 37, 46]. Thus control of nonlinear systems with nonlinear control inputs has received increasing attention [7, 22, 41, 46, 51]. However, the stability of FONNs cannot
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be established by using polynomial criteria [16, 29] because it is impossible to check the stability of FONNs simply by looking for its dominant roots. In [39],
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an adaptive switching control method was proposed for fractional-order systems (FOSs) subject to input nonlinearities, but the method used in the stability analysis is the integer-order Lyapunov method. Therefore, a fractional-order analysis method for stability of FONNs with nonlinear inputs is expected.
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The present paper will design an AFC for FONNs subject to input nonlinearities and dead-zones. There are three reasons that motivate us to study this
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control problem. Firstly, since system models are partly known for most works about the control and synchronization of FONNs, it is advisable to develop control methods for FONNs with fully unknown system structures. Secondly, few
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results have been given for discussion of the control of FOSs with input nonlinearities. Thirdly, the fractional-derivative of the quadratic Lyapunov function V (t) = 12 eT (t)P e(t) (which is usually be utilized to discuss the system’s stability) is given by the following complex form: C α 0 Dt V
(t) =
∞ X
k=0
Γ(1 + α) α−k C k D e(t)C e(t). 0 Dt Γ(1 + k)Γ(1 − k + α) 0 t 2
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Clearly, it is difficult to verify the stability of the FOSs by using this Lyapunov function [40, 43]. Therefore, new ideas or methods are needed. Compared with existing results (for examples, [1, 19, 39]), our study has the following contributions:
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1) An AFC is proposed for uncertain FONNs with fully unknown system
structures, where the new results (see Lemmas 4 and 5) can be expediently used to check the stability of FONNs by using fractional Lyapunov methods.
2) Input nonlinearities with unknown gain reduction tolerances are considered, and the values of gain reduction tolerances and the upper bound of unknown nonlinear functions are estimated by FLSs.
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3) A FOAL is designed to estimate adjustable parameter of the FLS online. The stability of the closed-loop system is proved strictly by using fractional Lyapunov stability criterion, where the asymptotic convergence of system variables can be guaranteed. Moreover, some controllers which are valid for NNs can be extended to control FONNs based on our results.
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2. Preliminaries and problem statement 2.1. Fractional calculus
The α-th fractional integral and α-th fractional derivative are defined by the
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following equalities (1) and (2)
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−α 0 Dt f (t) =
C α 0 Dt f (t)
=
1 Γ(α)
Z
t
(t − ξ)α−1 f (ξ)dξ,
0
1 Γ(1 − α)
Z
0
t
(t − ξ)−α f 0 (ξ)dξ,
(1)
(2)
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respectively, where Γ is the ordinary Gamma function. It follows from (2) that L
C α 0 Dt f (t)
= sα F (s) −
n−1 X
sα−k−1 f (k) (0)
(3)
k=0
where F (s) = L {f (t)}.
Definition 1. [38]. The mapping Eα,γ : C −→ C, defined by Eα,γ (ζ) =
∞ X
k=0
3
ζk , Γ(αk + γ)
(4)
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is called Mittag-Leffler function, where C is the set of all complex numbers, and α, γ > 0. Taking Laplace transform on (4) gives [38] sα−γ . sα + a
(5)
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L {tγ−1 Eα,γ (−atα )} =
Definition 2. [19]. A strictly increasing function h : [0, t) −→ [0, ∞) satisfying h(0) = 0 is called a class-K function.
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Definition 3. [19] A continuous function z(t) ∈ Rn (i.e. z : [0, ∞) −→ Rn ) is said to be Mittag-Leffler stable if the Euclidean norm of z(t) satisfies b
kz(t)k ≤ {g[z(0)]Eα (−κtα )} ,
(6)
where κ > 0, b > 0, g : Rn −→ [0, ∞) is locally Lipschitz on {z(t) (i.e. on {z(t) | t ∈ [0, ∞)}) and also satisfies g(00) = 0, and 0 = [0, · · · , 0]T ∈ Rn .
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Lemma 1. [19] Suppose that
C α 0 Dt h(t)
= f (t, h(t))
(7)
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with 0 being an equilibrium point. If there exist some class-K functions g1 , g2 , g3 and a Lyapunov function V (t, h(t)) such that
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g1 (kh(t)k) ≤ V (t, h(t)) ≤ g2 (kh(t)k), C α 0 Dt V
(t, h(t)) ≤ −g3 (kh(t)k),
(8) (9)
then lim h(t) = 0. t→∞
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Lemma 2. [27] If z(t) ∈ Rn is a smooth function, then 1C α T α D z (t)Bz(t) ≤ z T (t)B C 0 Dt z(t), 20 t
where B ∈ Rn×n is positive definite.
Lemma 3. If
C α 0 Dt z(t)
≤ 0, then z(t) ≤ z(0) (t ∈ [0, +∞)).
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(10)
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α Proof. Let h(t) = −C 0 Dt z(t) (t ∈ [0, +∞)). Then, for each t ∈ [0, +∞),
h(t) ≥ 0 and
C α 0 Dt z(t)
+ h(t) = 0.
(11)
Z(s) =
z(0) H(s) − α . s s
From (12) one has −α z(t) = z(0) − C 0 Dt h(t),
which implies z(t) ≤ x(0) (t ∈ [0, +∞)).
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Lemma 4. Let V1 (t) = 12 x2 (t) + 12 y 2 (t). If
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It follows from (11) that
C α 0 Dt V1 (t)
where k > 0. Then
≤ −kx2 (t),
x2 (t) ≤ 2V1 (0)Eα (−2ktα ). Proof. Using the operator
C −α 0 Dt
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−α 2 x2 (t) ≤ 2V1 (0) − 2k C 0 Dt x (t),
and thus
(13)
(14) (15)
to both sides of (14) gives
−α 2 V1 (t) − V1 (0) ≤ −k C 0 Dt x (t).
By (16),
(12)
−α 2 x2 (t) + m(t) = 2V1 (0) − 2k C 0 Dt x (t),
(16)
(17)
(18)
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where m(t) ≥ 0. It follows from (18) that X2 (s) = 2V1 (0)
sα−1 sα −2 α M (s), + 2k s + 2k
sα
(19)
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where X2 (s) = L {x2 (t)}. According to (5), (19) can be solved as x2 (t) = 2V1 (0)Eα (−2ktα ) − 2m(t) ∗ [t−1 Eα,0 (−2ktα )],
(20)
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where ∗ is the convolution operator. Thus Eα,0 (−2ktα ) ≥ 0. By (20), we know that (15) holds.
Lemma 5. Let V2 (t) = 21 xT (t)Q1 x(t)+ 12 z T (t)Q2 z(t), where x(t) and z(t) ∈ Rn are smooth functions, and Q1 , Q2 ∈ Rn×n are two positive definite matrices. If C α 0 Dt V2 (t)
≤ −h0 xT (t)Q3 x(t),
where Q3 > 0, h0 > 0, then lim kx(t)k = 0. t→∞
5
(21)
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2.2. Description of FLS
j∈J
(22)
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Usually, a FLS is modeled by [7, 8, 20, 21, 50]: P θj (t)µj (x(t)) j∈J P , fˆ(x(t)) = µj (x(t))
where fˆ (a Lipschitz continuous mapping from a compact subset Ω ⊆ Rn to n Q the real line R) is called the output, x is called the input, J = Fi , Fi i=1
consists of Ni fuzzy numbers (1 ≤ i ≤ n), µj (a mapping from Rn to the
closed unit interval [0, 1] ⊆ R which is defined relying on the above fuzzy numbers) is called the membership function of rule j
(j ∈ J), and θj is
We may identify
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called the centroid of the j-th consequent set (j ∈ J).
J with {1, 2, · · · , N } for convenience. Denote θ(t) = [θ1 (t), · · · , θN (t)]T and ψ(x(t)) = [q1 (x(t)), q2 (x(t)), · · · , qN (x(t))]T , where qj (called the j-th fuzzy ba-
sis function, j ∈ J) is a continuous mapping (and thus, ψ : Ω −→ RN is continuous) defined by
θ (t) qj (x(t)) = P j . µs (x(t))
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s∈J
Consequently, system (22) can be rewritten as
(23)
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fˆ(x(t)) = θT (t)ψ(x(t)).
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2.3. Problem statements
Consider the following FONNs: = −ci xi (t)+
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C α 0 Dt xi (t)
n X
aij fj (xj (t))+Ii +
j=1
n X j=1
gij ϕj (uj (t)), i = 1, · · · , n, (24)
in which n is the number of units, xi (t) represents the i-th unit’s state, aij
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corresponds to the constant connection weight, ci is the rate, gij is the unknown constant control gain, ϕj (uj (t)) represents a nonlinear input function, and Ii corresponds to the external input. Suppose that fi is Lipschitz-continuous.
Thus, system (24) has an unique solution [15]. Let x(t) = [x1 (t), · · · , xn (t)]T , ϕ(u(t)) = [ϕ1 (u1 (t)), · · · , ϕn (un (t))]T , C =
diag(c1 , · · · , cn ), I = [I1 , · · · , In ]T ,
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g11 · · · g1n . .. .. . G= . . . gn1 · · · gnn Then FONN (24) can be C α 0 Dt x(t)
a11 . , A = . . an1 rewritten as
··· .. . ···
a1n .. . . ann
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= −Cx(t) + Gϕ(u(t)) + Af (x(t)) + I.
(25)
Assumption 1. The unknown control gain matrix G in system (25) is positive definite. 2.4. Description of the input nonlinearities
The input nonlinearities usually exist due to the limitations of actuators.
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The model of ϕi (ui (t)) is defined as in [7, 39]: hi+ (ui (t))(ui (t) − µ∗+ ), ui (t) > µ∗+ , ϕi (ui (t)) = 0, − µ∗− ≤ ui (t) ≤ µ∗+ , h (u (t))(u (t) + µ∗ ), u (t) < −µ∗ , i− i i i − −
(26)
where hi+ (ui (t)) > 0 and hi− (ui (t)) > 0 are nonlinear functions, and µ∗+ and
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µ∗− are known constants. Based on the results in [7] and [39], it is reasonable to assume that ϕi (ui (t)) satisfies the followings:
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(ui (t) − µ∗+ )ϕi (ui (t)) ≥ d∗i+ (ui (t) − µ∗+ )2 ,
(ui (t) + µ∗− )ϕi (ui (t)) ≥ d∗i− (ui (t) + µ∗− )2 ,
(27)
where d∗i+ and d∗i− are “gain reduction tolerances”. Denote di = min{d∗i+ , d∗i− }.
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A dead-zone input nonlinearity is presented in Figure 1.
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Assumption 2. The input nonlinearity ϕi (ui (t)) is unknown, and the gain reduction tolerances, d∗i+ and d∗i− , are also unknown (furthermore, di is unknown).
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Remark 1. In fact, the model (26) was also used in [12, 31, 32, 39]. However, in above literature, the exact values of gain reduction tolerances should be known in advance. 3. Design of AFC and stability analysis Denote G−1 = Λ. As G > 0, Λ > 0. Multiplying both sides of (25) by Λ
gives α ΛC 0 Dt x(t) = −ΛCx(t) + ϕ(u(t)) + ΛAf (x(t)) + ΛI.
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(28)
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Note that f (x(t)) and Λ are unknown, we can rewrite (28) into the following from: α ΛC 0 Dt x(t) = −ΛCx(t) + $(t) + ϕ(u(t)),
(29)
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where $(t) = ΛAf (x(t)) + ΛI is an unknown nonlinear function. It is easy to know that $(x(t)) = ΛAf (x(t)) + ΛI is a Lipschitz continuous function, thus we can give the following assumption.
Assumption 3. There exists a continuous function $ ¯ i (x(t)) such that |$i (x(t))| ≤ d$ ¯ i (x(t)), where d = min {di }.
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1≤i≤n
(30)
Remark 2. Assumption 3 is not restrictive because the upper bound function $ ¯ i (x(t)) is assumed to be unknown. At the same time, $ ¯ i (x(t)) always exists since $(x(t)) is continuous. We can employ FLS (23) to approximate $ ¯ i (t) by
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ˆ $ ¯ i (t) = θiT (t)ψi (t),
(31)
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where ψi (t) is a fuzzy basis function, and θi (t) is an adjustable parameter of the ˆ ¯ i (t) − $ ¯ i (t) } as small as possible. Let the parameter FLS, which makes sup{ $ estimation error be
θ˜i (t) = θi (t) − θi∗ ,
(32)
ˆ εi (t) = $ ¯ i (t) − $ ¯ i (t).
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and let the fuzzy approximation error be (33)
As in the literature [7, 33, 47], it is reasonable to assume that sup{|εi (x(t))|} ≤
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ε∗ for some ε∗ > 0. Consequently, we have
ˆ ˆ ˆ $ ¯ i (t) − $i (t) = $ ¯ i (θi (t), x(t)) − $ ¯ i (t) − εi (t) = θ˜iT (t)ψi (t) − εi (t).
Thus the controller can constructed as following: − uri (t)sign(xi (t)) − µ∗− , xi (t) > 0, ui (t) = 0, xi (t) = 0, − u (t)sign(x (t)) + µ∗ , x (t) < 0, ri i i + 8
(34)
(35)
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with uri = b + θiT (t)ψi (x(t)),
(36)
where b > ∗ .
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Remark 3. Owing to the special form of FONNs (see the term −Cx(t) in (24)), the proposed controller (35) is different from that by conventional AFC methods for integer-order nonlinear systems (see [7]). Moreover, one dose not need to know the sign of the system states.
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Remark 4. Although the condition b ∈ (ε∗ , +∞) is needed in the controller design, it is actually difficult to know the exact value of ε∗ . On the other hand, we may see in section 4 that good control performance can be obtained even if b is small (b = 1), which implies that the proposed control method has good robustness. The FOAL is given as C α 0 Dt θi (t)
= γi |xi (t)|ψi (x(t)), θi (0) ≥ 0,
(37)
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where γi > 0 is the design parameter.
Remark 5. FOAL (37) can also be expressed as:
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1 θi (t) = θi (0) + Γ(α)
Z
0
t
$i (t − ξ)α−1 |xi (ξ)|ψi (x(ξ))dξ.
(38)
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Remark 6. As the term γi |xi (t)|ψi (x(t)) in (37) is not negative, θ(t) ≥ 0 by Lemma 3. As a result, we have uri ≥ 0.
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Theorem 1. Consider the FONN (25) under Assumptions 1-3. The AFC (35) together with the FOAL (37) ensures that lim xi (t) = 0. t→∞
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Proof. Multiplying xT (t) to both sides of (29) we have α T T T xT (t)ΛC 0 Dt x(t) = x (t)$(t) − x (t)ΛCx(t) + x (t)ϕ(u(t))
= −xT (t)ΛCx(t) + ≤ −xT (t)ΛCx(t) +
n X
k=1 n X
k=1
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xk (t)$k (t) + xT (t)ϕ(u(t)) ¯ k (t) + xT (t)ϕ(u(t)). |xk (t)|d$
(39)
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By (34), we obtain n X
k=1
|xk (t)|d$ ¯ k (x(t)) + xT (t)ϕ(u(t)) − xT (t)ΛCx(t)
= xT (t)ϕ(u(t)) − xT (t)ΛCx(t) + d −d
n X
k=1
|xk (t)|θ˜kT (t)ψk (t) + d
n X
k=1
n X
k=1
|xk (t)|εk (t)
≤ −xT (t)ΛCx(t) + xT (t)ϕ(u(t)) + d
k=1
|xk (t)| − d
n X
n X
k=1
|xk (t)|θkT (t)ψk (t)
|xk (t)|θ˜kT (t)ψk (t).
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+ dε∗
n X
|xk (t)|θkT (t)ψk (x(t))
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α xT (t)ΛC 0 Dt x(t) ≤
k=1
(40)
From (35) we know ui ≤ −µ∗− for xi (t) > 0 and ui > µ∗+ for xi (t) ≤ 0. By
(27), we have
(ui (t) + µ∗− )ϕi (ui (t)) ≥ d∗i− u2ri (t) ≥ du2ri (t)
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for xi (t) > 0, and
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(ui (t) − µ∗+ )ϕi (ui (t)) ≥ d∗i+ u2ri (t) ≥ du2ri (t)
(41)
(42)
for xi (t) ≤ 0. Consequently,
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− uri (t)x2i (t)sign(xi (t))ϕi (ui (t)) ≥ du2ri (t)x2i (t) = du2ri (t)|xi (t)|2 .
(43)
xi (t)ϕi (ui (t)) ≤ −duri (t)|xi (t)|.
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That means, for all xi (t),
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(44)
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Substituting (36) and (44) into (40) gives α T xT (t)ΛC 0 Dt x(t) ≤ −duri (t)|xi (t)| − x (t)ΛCx(t) − d
k=1
|xk (t)| + d
= −xT (t)ΛCx(t) − d + d(ε∗ − b)
n X
k=1
n X
k=1
n X
k=1
|xk (t)|θkT (t)ψi (t)
|xk (t)|θ˜kT (t)ψi (t)
|xk (t)|.
(45)
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If we set the design parameter b > ε∗ , then
α T xT (t)ΛC 0 Dt x(t) ≤ −x (t)ΛCx(t) − d
Define
k=1
|xk (t)|θ˜kT (t)ψk (t)
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+ dε∗
n X
n X
n X
k=1
|xk (t)|θ˜kT (t)ψk (t).
(46)
n
ν(t) =
1 T 1 X 1 ˜T ˜ x (t)Λx(t) + θ (t)θk (t). 2d 2 γk k
(47)
k=1
≤
n X 1 ˜T C α ˜ 1 T α x (t)ΛC D x(t) + θ (t)0 Dt θi (t). 0 t d γ i i=1 i
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C α 0 Dt ν(t)
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According to Lemma 2, it holds that
(48)
From (32) we have
α =C 0 Dt θi (t).
(49)
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C α˜ 0 Dt θi (t)
Substituting (37), (46) and (49) into (48), after some straightforward ma-
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nipulators, we have
α 0 Dt V
1 (t) ≤ − xT (t)ΛCx(t). d
(50)
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From (50) and Lemma 5, we know that both xi (t) and θ˜i (t) are bounded. Consequently, θi (t) is bounded. Because uri is bounded by (36), ui (t) is bounded.
Note that ΛC > 0, xi (t) converges to 0 asymptotically (i.e., lim xi (t) = 0). t→∞
Remark 7. (1) AFC method is also used to control FOSs in [23], where the α-th q T derivative of a squared Lyapunov 21 eT (t)Λe(t) is given by 21 (C 0 Dt e(t)) Λe(t) +
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1 C q 2 e(t)Λ(0 Dt e(t))
(see Equality (33)). However, this is incorrect, the correct
form is
α T T C α = (C 0 Dt e(t)) e(t) + e (t)0 Dt e(t) ∞ X Γ(1 + α) α−k C k +2 D e(t)C e(t). 0 Dt Γ(1 + k)Γ(1 − k + α) 0 t
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C α T 0 Dt 2e (t)e(t)
k=1
(2) Actually, we need not to handle the above complicated infinite series. As shown in Theorem 1, we give a strict proof of the stability of the FONNs based on Lemmas 1 and 2 and the proposed Lemmas 3, 4 and 5.
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Remark 8. In [39], adaptive controller has been proposed for FOSs with input nonlinearities. The comparisons between the results of this paper and that of [39] is included in Table 1.
4. Simulation results
In this section three simulation examples will be given to demonstrate the
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effectiveness of the proposed method. 4.1. Example 1
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In system (24), let n = 3, α = 0.97, fi (xi (t)) = tanh(xi (t)), ci = 1, and
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Ii = 0 (i = 1, 2, 3). 2.00 −1.20 0 0.5 0 0 A= 1.71 1.15 2.00 , G = 0 0.5 0 . −4.75 0 1.10 0 0 0.7 The input nonlinearities ϕi (ui (t)) (i = 1, 3) are given as: (ui (t) − 2)(1 − 0.3 sin(ui (t))), ui (t) > 2.0, ϕi (ui (t)) = 0, − 2.0 ≤ ui (t) ≤ 2.0, (u (t) + 2)(0.8 − 0.3 cos(u (t))), u (t) < −2.0. i
i
i
The input nonlinearity ϕ2 (u2 (t)) is chosen as ui (t) > 5, (1 − 0.3 sin(u2 (t)))(ui (t) − 5), ϕ2 (u2 (t)) = 0, − 5 ≤ ui (t) ≤ 5, (0.8 − 0.3 cos(u (t)))(u (t) + 5), u (t) < −5. i i i 12
(51)
(52)
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Table 1: Our results and the results in [39].
The results in [39]
Our results
System model
MIMO FOSs, and prior knowledge
MIMO FONNs, and the structure
should be known
can be fully unknown
Control method
Switching adaptive controller
AFC
Gain reduction toler-
Their
be
Their exact values are not needed
ances
known in advance
Design parameters
There are 3n control parameters to
There are n + 1 control parameters
be chosen
to be chosen
Parameters to be up-
There are 3n parameters that
There are nN parameters that
dated
should be updated online
should be updated
The advantages of the control
The advantages of our method: (1)
methods in [39]: (1) fewer param-
better robustness; (2) the exact
eters to be updated, less computa-
values of the gain reduction toler-
tion burden; (2) the controller is
ances can be unknown (it is dif-
easier to design and may be more
ficult to obtain their exact values
convenient for practical use
[7])
should
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values
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Conclusions
exact
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Comparison
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Under these parameters and without the control input (i.e., ui (t) ≡ 0), the
dynamical behavior of system (24) is shown in Figure 2.
3
4
2 x2(t)
x3(t)
2 0
x2(t)
0
−3
−2
−1 x1(t)
(c)
6
−1
1
0
−2 −4
−3
−2
(d)
6
0
1
4
x3(t)
4
−1 x1(t)
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2
x3(t)
1 0
4
2 0
2 0
−3
−2
−1 x1(t)
0
−2 −2
1
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−2 −4
(b)
4
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(a)
0
2
4
x2(t)
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Figure 2: Dynamical behavior of system (24) in (a) x1 (t) − x2 (t) − x3 (t), (b) x1 (t) − x2 (t) plane, (c) x1 (t) − x3 (t) plane and (d) x1 (t) − x4 (t) plane.
It is worth mentioning, within all simulation studies, the proposed con-
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trol method does not need the knowledge of the FONNs’s model. Indeed, the FONNs’s model is only given for simulation purposes. The FLSs use x1 (t), x2 (t)
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and x3 (t) as their inputs. From Figure 2 we know that the system variables distribute on interval [-3, 3]. Thus, for each input variable, we define 3 Gaussian membership functions distribute uniformly on interval [-3, 3]. As a result, there are 3 × 3 × 3 = 27 fuzzy rules are used in the simulation. The Gaussian mem-
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bership functions are given in Figure 3. The design parameters are chosen as b = 1, γ1 = γ2 = γ3 = 10, x(0) = [2.0, −2.0, 3.0]T , θi (0) = 0 ∈ R27 .
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1
0.6
0.4
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Membership functions
0.8
0.2
0 −3
−2
−1
0 xi(t),i=1,2,3
1
2
3
Figure 3: Membership functions.
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In the simulation, sign(·) is replaced by arctan(10·). Our results are given in Figures 4, 5, and 6. These results indicate that good control performance has been achieved. Figure 4 gives the state variables. Figure 5 presents the control inputs. Noting that the input nonlinearities are given as (51) and (52), the control input u2 (t) tends to 5, and ui (t), i = 1, 3 tend to -2 eventually. Figure 6 depicts the input nonlinearities. It should be mentioned that, although b is set very small, the states converge to 0 rapidly. That is to say that the employed
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FLSs have good approximation abilities.
x1(t)
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3
x2(t) x3(t)
1
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State variables
2
0
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−1
−2
0
1
2
3
4
5 6 Time(second)
7
Figure 4: The state variables.
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10 u1(t) u2(t) u3(t)
0
−5
−10
0
1
2
3
4
5 6 Time(second)
7
8
6
Φ2(u2(t)) Φ3(u3(t))
2 0 −2
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Input nonlinearities
10
Φ1(u1(t))
4
−4 −6
0
1
2
3
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−8
9
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Figure 5: Control inputs.
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Control inputs
5
4
5 6 Time(second)
7
8
9
10
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Figure 6: The input nonlinearities.
In the remainder of this subsection, we will show how different choices of membership functions and fuzzy rules affect the control performance. We give
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four examples presented in Table 2 (the step is 0.02) by using different membership functions and fuzzy rules. The simulation results are included in Figure 7. In this paper, the fuzzy membership functions are chosen according to expertise
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and control performance. It should be mentioned that similar functions are used in [6, 7, 21, 22, 23, 26, 33, 45, 46]. According to simulation results in Figure 7, the proposed AFC works well even when different membership functions are used. It can be concluded that, in all these cases, the simulation results are similar. However, when more fuzzy membership functions are chosen, more fuzzy rules and simulation time are needed. Moreover, the triangular membership function
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computation will consume more time than the Gaussian function. Thus, in the beginning of this subsection, we use three Gaussian functions in the simulation. Further investigation is needed to determine the optimal number of membership
Table 2: Different membership functions
Member functions
Parameters
Fuzzy rules
Time consumed
3 Gaussian functions
[1.2, i], i = 1, 2, 3
27
1.52 seconds
5 Gaussian functions
[0.4, i], i = 1, · · · , 5
125
5 Triangular functions
[i − 2, i, i + 2], i = 1, · · · , 5
125
[0.2, i], i = 1, · · · , 7
343
Case 1: 3 Gaussian functions
4
0
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2
x1(t) x3(t)
0
0
x3(t)
0 −1 0
5
10
Case 4: 7 Gaussian functions x1(t)
3
x2(t)
−1 −2
x2(t)
1
4
1
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State variables
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3
x1(t)
Time(second)
Case 3: 5 triangular functions
4
2
−2
10
State variables
−2
State variables
x3(t)
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State variables
2
−1
51.63 seconds
3
x2(t)
0
10.63 seconds
Case 2: 5 Gaussian functions
4
x1(t)
3
1
7.88 seconds
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7 Gaussian functions
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functions and their parameters to achieve better control performance.
x2(t)
2
x3(t)
1 0 −1
5 Time(second)
−2
10
0
5
10
Time(second)
Figure 7: Simulation results for different choices of membership functions.
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4.2. Example 2 Suppose that the FONNs have time-varying gain aij (t) (i.e., the matrix A is replaced by A(t)) and external disturbances [5], which is more complicated
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than above model: α = 0.98, fi (xi (t)) = tanh(|x i (t)| − 1), ci = 1, Ii = 0, a11 (x1 (t)) 1 −9 1.99, |xi (t)| ≤ 1, , aii (xi (t)) = A= −9 a22 (x2 (t)) 1 −1.99, |x (t)| > 1. i 1 −9 a33 (x3 (t)) To show the robustness of the control system, let us add three external distur-
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bances to system (24): d1 (t) = 0.48 sin t, d2 (t) = −0.48 cos t, d3 (t) = 0.49 sin t − 1.2 −0.2 0.1 0.5 cos t. The control gain matrix is chosen as G = −0.2 1.2 0.1 , which 0.1 0.1 1.7 is a positive definite matrix. The FLSs has control parameters and input nonlinearities chosen as above. Let x(0) = [−3, 2, −3] and θi (0) = 0 ∈ R27 , i = 1, 2, 3. Our results are presented in Figures 8 and 9. From Figure 8 we can see that
the state variables converge to the origin very fast, and have small fluctuations. Figure 9 depicts the time response of the input nonlinearities. The qualitative
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analysis of these results is the same as that of Example 1.
1 0
x2(t) x3(t)
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State variables
x1(t)
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2
−1
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−2 −3
AC
−4
0
1
2
3
4
5 6 Time(second)
Figure 8: The state variables.
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7
8
9
10
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5 Φ1(u1(t))
0
−5
0
1
2
3
4
5 6 Time(second)
7
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Φ3(u3(t))
8
9
10
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Input nonlinearities
Φ2(u2(t))
Figure 9: The input nonlinearities.
4.3. Example 3
It should be mentioned that our controller is also valid for many classes of
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FOSs. We will give a comparison between our control method and that of [39]. The mathematical model of the uncertain FOS used in [39] (see Equality (76)
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on page 218 in [39]) can be described as C α 0 Dt x1 (t) =10(x2 (t) − x1 (t)) + x4 (t) + 0.2 cos(3x1 (t))x1 (t) + ϕj (u1 (t)), α C 0 Dt x2 (t) =28x1 (t) − x2 (t) + x1 (t)x3 (t) + 0.12 sin(4x2 (t))x2 (t) + ϕ2 (u2 (t)), (53)
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8 C α 0 Dt x3 (t) =x1 (t)x2 (t) − x3 (t) + 0.12 sin(4x2 (t))x2 (t) + ϕ3 (u3 (t)), 3 C Dα x (t) = − x (t)x (t) − x (t) + 0.1 cos(2x (t))x (t) + ϕ (u (t)), 2 3 4 4 4 4 4 0 t 4
with α = 0.93.
Let the initial condition be x(0) = [2, 4, 1, 3]T . The input nonlinearities
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ϕi (ui (t)) (i = 1, 3) are given as (51). The control parameters in [39] are chosen as ci = 0.25, ηi = 8.24, κi λi = 6 (see Equalities (78) and (79) in [39]). Our control parameters are chosen the same as that in Example 2. The simulation results are presented in Figures 10 and 11. The simulation results indicate that under the chosen parameters, compared with the control method in [39], our method has smaller control inputs and faster convergence of the state vector.
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In (53), d(t) = [0.2x1 cos 3x1 , 0.12x2 sin 4x2 , 0.12x2 sin 4x2 , 0.1x4 cos 2x4 ]T . To show the robustness of the proposed control method, let us use larger disturbance: cd(t) where c is a positive constant. The simulation results are shown in Figure 12. It can be concluded that by using the control method in [39],
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when the disturbance is chosen as 3d(t), the system variables needs more time
to converge to the origin. When the disturbance is chosen as 6d(t), the controller proposed by [39] will be not valid anymore (to overcome this problem, one can choose larger control parameters). However, our control method works
well when the disturbance changes largely. That is to say, the proposed AFC
(a)
x 2 (t) x 3 (t) x 4 (t)
2 0
0.5
1
1.5
Time(second) (c)
6
x 1 (t)
4 2
0
0.5
1
1.5
x 3 (t) x 4 (t)
2 1
0
Time(second)
x 2 (t) x 3 (t) x 4 (t)
0.4
0.6
x 1 (t)
4
x 2 (t) x 3 (t)
2
x 4 (t)
0 -2
2
0.2
x 1 (t)
Time(second) (d)
6
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0
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x 2 (t)
(b)
×10 284
3
0
2
State variables
4
State variables
x 1 (t)
6
0
State variables
4
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State variables
8
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method has satisfactory robustness.
0
0.5
1
1.5
2
Time(second)
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Figure 12: Our results and the results in [39] with larger disturbance: (a) the results of [39] when c = 3; (b) the results of [39] when c = 6; (c) our results when c = 3; (d) our results
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when c = 6.
5. Conclusions An AFC for a class of FONNs with input nonlinearities and dead-zones has
been proposed in the paper, where the unknown structure of the FONNs and the exact values of gain reduction tolerances are approximated by FLSs. By using the proposed FOAL, the stability of the closed-loop system is proven strictly 20
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based on fractional Lyapunov criteria. By using the results of this study, many control methods that are valid for integer-order NNs can be extended to control FONNs. Simulation studies have demonstrated that the proposed method is not only efficient for obtaining parameter convergence but also useful for improving
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the control performance and robustness. Future work will be focused on how
to choose appropriate fuzzy rules and membership functions to achieve better control performance. 6. Acknowledgements
This work is supported by the National Natural Science Foundation of China
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under Grant 11771263, and the Natural Science Foundation of Anhui Province
of China under Grant 1808085MF181. The authors would like to thank the editor and the anonymous referees for their comments and suggestions. References
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Figure 1: Input nonlinearity.
(a)
3
(b)
6
x 2 (t)
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4
x 1 (t)
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0
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1
Time(second) (c)
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Figure 10: The time response of the state variables under our control approach (solid line)
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and the method of [39] (dashed line) in (a) x1 (t); (b) x2 (t); (c) x3 (t); (d) x4 (t).
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(a)
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(b)
100
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Time(second) (c)
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Time(second)
2
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-50
2
1
Time(second) (d)
50
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20
-60
1.5
u 4 (t)
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Figure 11: The time response of the control inputs under our control approach (solid line)
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and the method of [39] (dashed line) in (a) u1 (t); (b) u2 (t); (c) u3 (t); (d) u4 (t).
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Adaptive Fuzzy Control for a Class of Unknown Fractional-Order Neural Networks Subject to Input Nonlinearities and Dead-Zones Heng Liu, Shenggang Li, Hongxing Wang, Yeguo Sun PII: DOI: Reference:
S0020-0255(18)30326-8 10.1016/j.ins.2018.04.069 INS 13612
To appear in:
Information Sciences
Received date: Revised date: Accepted date:
6 March 2017 13 April 2018 21 April 2018
Please cite this article as: Heng Liu, Shenggang Li, Hongxing Wang, Yeguo Sun, Adaptive Fuzzy Control for a Class of Unknown Fractional-Order Neural Networks Subject to Input Nonlinearities and Dead-Zones, Information Sciences (2018), doi: 10.1016/j.ins.2018.04.069
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Adaptive Fuzzy Control for a Class of Unknown Fractional-Order Neural Networks Subject to Input Nonlinearities and Dead-Zones
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Heng Liua,b , Shenggang Lia,∗, Hongxing Wangb , Yeguo Sunb a College
of Mathematics and Information Science, Shaanxi Normal Universtiy, Xi’an 710119, China b Department of Mathematics and Computational Science, Huainan Normal University, Huainan 232038, China
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Abstract
This paper presents an adaptive fuzzy control (AFC) for uncertain fractionalorder neural networks (FONNs) with input nonlinearities and unmodeled dynamics. System uncertainties and unknown parts of the nonlinear input are approximated by fuzzy logic systems (FLSs). Based on some proposed stability analysis criteria for fractional-order systems (FOSs), an AFC is designed to guar-
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antee the asymptotic stability of the controlled system. Fractional-order adaptive laws (FOALs) are constructed to update adjustable parameters of FLSs.
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Our method can be used to control FONNs with/without sector nonlinearities in control inputs. It also allows us to generalize many existing control methods that are valid for integer-order neural networks to FONNs by using the proposed
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method. Finally, the effectiveness of the proposed method is demonstrated by simulation results.
Keywords: Adaptive fuzzy control, sector nonlinearity, fractional-order neural
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network, dead-zone
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1. Introduction Up to now, neural networks (NNs) have received increasing attention in
many fields [2, 13, 17, 28, 29, 34, 35, 45]. The NN that will be considered in this paper is called fractional-order neural network (FONN), which is an extension of ∗ Corresponding author, College of Mathematics and Information Science, Shaanxi Normal Universtiy, Xi’an 710119, China; Email: [email protected]
Preprint submitted to Elsevier
April 23, 2018
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ordinary NNs and have also been well studied [10, 28, 53] because the fractionalorder derivative provides a powerful technique for describing hereditary and memory properties of the system. FONNs have two advantages: one is that the fractional calculus has infinite memory, the other is that it has one more on the control and synchronization of FONNs.
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freedom [3, 27]. Please see [10, 14] and some references therein for recent results Fuzzy logic systems (FLSs) with linguistic information have been effectively
utilized in the control of unknown nonlinear systems [4, 6, 7, 9, 11, 18, 20, 21, 24,
25, 26, 30, 33, 36, 42, 47, 48, 49, 50]. However, there are few works that studied the adaptive fuzzy control (AFC) of FONNs because it is a challenging work to
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check the stability by using a fractional-order stability criterion. On the other
hand, we notice that inputs (particularly, nonlinear inputs), outputs, and state constraints, which are reflected by physical stoppages, dead-zones, backlash, saturation, and safety specifications, are very common in most physical systems, and nonlinear inputs usually lead to instability of the controlled system [7, 37, 46]. Thus control of nonlinear systems with nonlinear control inputs has received increasing attention [7, 22, 41, 46, 51]. However, the stability of FONNs cannot
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be established by using polynomial criteria [16, 29] because it is impossible to check the stability of FONNs simply by looking for its dominant roots. In [39],
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an adaptive switching control method was proposed for fractional-order systems (FOSs) subject to input nonlinearities, but the method used in the stability analysis is the integer-order Lyapunov method. Therefore, a fractional-order analysis method for stability of FONNs with nonlinear inputs is expected.
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The present paper will design an AFC for FONNs subject to input nonlinearities and dead-zones. There are three reasons that motivate us to study this
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control problem. Firstly, since system models are partly known for most works about the control and synchronization of FONNs, it is advisable to develop control methods for FONNs with fully unknown system structures. Secondly, few
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results have been given for discussion of the control of FOSs with input nonlinearities. Thirdly, the fractional-derivative of the quadratic Lyapunov function V (t) = 12 eT (t)P e(t) (which is usually be utilized to discuss the system’s stability) is given by the following complex form: C α 0 Dt V
(t) =
∞ X
k=0
Γ(1 + α) α−k C k D e(t)C e(t). 0 Dt Γ(1 + k)Γ(1 − k + α) 0 t 2
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Clearly, it is difficult to verify the stability of the FOSs by using this Lyapunov function [40, 43]. Therefore, new ideas or methods are needed. Compared with existing results (for examples, [1, 19, 39]), our study has the following contributions:
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1) An AFC is proposed for uncertain FONNs with fully unknown system
structures, where the new results (see Lemmas 4 and 5) can be expediently used to check the stability of FONNs by using fractional Lyapunov methods.
2) Input nonlinearities with unknown gain reduction tolerances are considered, and the values of gain reduction tolerances and the upper bound of unknown nonlinear functions are estimated by FLSs.
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3) A FOAL is designed to estimate adjustable parameter of the FLS online. The stability of the closed-loop system is proved strictly by using fractional Lyapunov stability criterion, where the asymptotic convergence of system variables can be guaranteed. Moreover, some controllers which are valid for NNs can be extended to control FONNs based on our results.
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2. Preliminaries and problem statement 2.1. Fractional calculus
The α-th fractional integral and α-th fractional derivative are defined by the
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following equalities (1) and (2)
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−α 0 Dt f (t) =
C α 0 Dt f (t)
=
1 Γ(α)
Z
t
(t − ξ)α−1 f (ξ)dξ,
0
1 Γ(1 − α)
Z
0
t
(t − ξ)−α f 0 (ξ)dξ,
(1)
(2)
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respectively, where Γ is the ordinary Gamma function. It follows from (2) that L
C α 0 Dt f (t)
= sα F (s) −
n−1 X
sα−k−1 f (k) (0)
(3)
k=0
where F (s) = L {f (t)}.
Definition 1. [38]. The mapping Eα,γ : C −→ C, defined by Eα,γ (ζ) =
∞ X
k=0
3
ζk , Γ(αk + γ)
(4)
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is called Mittag-Leffler function, where C is the set of all complex numbers, and α, γ > 0. Taking Laplace transform on (4) gives [38] sα−γ . sα + a
(5)
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L {tγ−1 Eα,γ (−atα )} =
Definition 2. [19]. A strictly increasing function h : [0, t) −→ [0, ∞) satisfying h(0) = 0 is called a class-K function.
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Definition 3. [19] A continuous function z(t) ∈ Rn (i.e. z : [0, ∞) −→ Rn ) is said to be Mittag-Leffler stable if the Euclidean norm of z(t) satisfies b
kz(t)k ≤ {g[z(0)]Eα (−κtα )} ,
(6)
where κ > 0, b > 0, g : Rn −→ [0, ∞) is locally Lipschitz on {z(t) (i.e. on {z(t) | t ∈ [0, ∞)}) and also satisfies g(00) = 0, and 0 = [0, · · · , 0]T ∈ Rn .
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Lemma 1. [19] Suppose that
C α 0 Dt h(t)
= f (t, h(t))
(7)
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with 0 being an equilibrium point. If there exist some class-K functions g1 , g2 , g3 and a Lyapunov function V (t, h(t)) such that
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g1 (kh(t)k) ≤ V (t, h(t)) ≤ g2 (kh(t)k), C α 0 Dt V
(t, h(t)) ≤ −g3 (kh(t)k),
(8) (9)
then lim h(t) = 0. t→∞
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Lemma 2. [27] If z(t) ∈ Rn is a smooth function, then 1C α T α D z (t)Bz(t) ≤ z T (t)B C 0 Dt z(t), 20 t
where B ∈ Rn×n is positive definite.
Lemma 3. If
C α 0 Dt z(t)
≤ 0, then z(t) ≤ z(0) (t ∈ [0, +∞)).
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(10)
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α Proof. Let h(t) = −C 0 Dt z(t) (t ∈ [0, +∞)). Then, for each t ∈ [0, +∞),
h(t) ≥ 0 and
C α 0 Dt z(t)
+ h(t) = 0.
(11)
Z(s) =
z(0) H(s) − α . s s
From (12) one has −α z(t) = z(0) − C 0 Dt h(t),
which implies z(t) ≤ x(0) (t ∈ [0, +∞)).
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Lemma 4. Let V1 (t) = 12 x2 (t) + 12 y 2 (t). If
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It follows from (11) that
C α 0 Dt V1 (t)
where k > 0. Then
≤ −kx2 (t),
x2 (t) ≤ 2V1 (0)Eα (−2ktα ). Proof. Using the operator
C −α 0 Dt
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−α 2 x2 (t) ≤ 2V1 (0) − 2k C 0 Dt x (t),
and thus
(13)
(14) (15)
to both sides of (14) gives
−α 2 V1 (t) − V1 (0) ≤ −k C 0 Dt x (t).
By (16),
(12)
−α 2 x2 (t) + m(t) = 2V1 (0) − 2k C 0 Dt x (t),
(16)
(17)
(18)
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where m(t) ≥ 0. It follows from (18) that X2 (s) = 2V1 (0)
sα−1 sα −2 α M (s), + 2k s + 2k
sα
(19)
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where X2 (s) = L {x2 (t)}. According to (5), (19) can be solved as x2 (t) = 2V1 (0)Eα (−2ktα ) − 2m(t) ∗ [t−1 Eα,0 (−2ktα )],
(20)
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where ∗ is the convolution operator. Thus Eα,0 (−2ktα ) ≥ 0. By (20), we know that (15) holds.
Lemma 5. Let V2 (t) = 21 xT (t)Q1 x(t)+ 12 z T (t)Q2 z(t), where x(t) and z(t) ∈ Rn are smooth functions, and Q1 , Q2 ∈ Rn×n are two positive definite matrices. If C α 0 Dt V2 (t)
≤ −h0 xT (t)Q3 x(t),
where Q3 > 0, h0 > 0, then lim kx(t)k = 0. t→∞
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(21)
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2.2. Description of FLS
j∈J
(22)
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Usually, a FLS is modeled by [7, 8, 20, 21, 50]: P θj (t)µj (x(t)) j∈J P , fˆ(x(t)) = µj (x(t))
where fˆ (a Lipschitz continuous mapping from a compact subset Ω ⊆ Rn to n Q the real line R) is called the output, x is called the input, J = Fi , Fi i=1
consists of Ni fuzzy numbers (1 ≤ i ≤ n), µj (a mapping from Rn to the
closed unit interval [0, 1] ⊆ R which is defined relying on the above fuzzy numbers) is called the membership function of rule j
(j ∈ J), and θj is
We may identify
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called the centroid of the j-th consequent set (j ∈ J).
J with {1, 2, · · · , N } for convenience. Denote θ(t) = [θ1 (t), · · · , θN (t)]T and ψ(x(t)) = [q1 (x(t)), q2 (x(t)), · · · , qN (x(t))]T , where qj (called the j-th fuzzy ba-
sis function, j ∈ J) is a continuous mapping (and thus, ψ : Ω −→ RN is continuous) defined by
θ (t) qj (x(t)) = P j . µs (x(t))
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s∈J
Consequently, system (22) can be rewritten as
(23)
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fˆ(x(t)) = θT (t)ψ(x(t)).
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2.3. Problem statements
Consider the following FONNs: = −ci xi (t)+
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C α 0 Dt xi (t)
n X
aij fj (xj (t))+Ii +
j=1
n X j=1
gij ϕj (uj (t)), i = 1, · · · , n, (24)
in which n is the number of units, xi (t) represents the i-th unit’s state, aij
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corresponds to the constant connection weight, ci is the rate, gij is the unknown constant control gain, ϕj (uj (t)) represents a nonlinear input function, and Ii corresponds to the external input. Suppose that fi is Lipschitz-continuous.
Thus, system (24) has an unique solution [15]. Let x(t) = [x1 (t), · · · , xn (t)]T , ϕ(u(t)) = [ϕ1 (u1 (t)), · · · , ϕn (un (t))]T , C =
diag(c1 , · · · , cn ), I = [I1 , · · · , In ]T ,
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g11 · · · g1n . .. .. . G= . . . gn1 · · · gnn Then FONN (24) can be C α 0 Dt x(t)
a11 . , A = . . an1 rewritten as
··· .. . ···
a1n .. . . ann
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= −Cx(t) + Gϕ(u(t)) + Af (x(t)) + I.
(25)
Assumption 1. The unknown control gain matrix G in system (25) is positive definite. 2.4. Description of the input nonlinearities
The input nonlinearities usually exist due to the limitations of actuators.
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The model of ϕi (ui (t)) is defined as in [7, 39]: hi+ (ui (t))(ui (t) − µ∗+ ), ui (t) > µ∗+ , ϕi (ui (t)) = 0, − µ∗− ≤ ui (t) ≤ µ∗+ , h (u (t))(u (t) + µ∗ ), u (t) < −µ∗ , i− i i i − −
(26)
where hi+ (ui (t)) > 0 and hi− (ui (t)) > 0 are nonlinear functions, and µ∗+ and
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µ∗− are known constants. Based on the results in [7] and [39], it is reasonable to assume that ϕi (ui (t)) satisfies the followings:
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(ui (t) − µ∗+ )ϕi (ui (t)) ≥ d∗i+ (ui (t) − µ∗+ )2 ,
(ui (t) + µ∗− )ϕi (ui (t)) ≥ d∗i− (ui (t) + µ∗− )2 ,
(27)
where d∗i+ and d∗i− are “gain reduction tolerances”. Denote di = min{d∗i+ , d∗i− }.
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A dead-zone input nonlinearity is presented in Figure 1.
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Assumption 2. The input nonlinearity ϕi (ui (t)) is unknown, and the gain reduction tolerances, d∗i+ and d∗i− , are also unknown (furthermore, di is unknown).
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Remark 1. In fact, the model (26) was also used in [12, 31, 32, 39]. However, in above literature, the exact values of gain reduction tolerances should be known in advance. 3. Design of AFC and stability analysis Denote G−1 = Λ. As G > 0, Λ > 0. Multiplying both sides of (25) by Λ
gives α ΛC 0 Dt x(t) = −ΛCx(t) + ϕ(u(t)) + ΛAf (x(t)) + ΛI.
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(28)
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Note that f (x(t)) and Λ are unknown, we can rewrite (28) into the following from: α ΛC 0 Dt x(t) = −ΛCx(t) + $(t) + ϕ(u(t)),
(29)
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where $(t) = ΛAf (x(t)) + ΛI is an unknown nonlinear function. It is easy to know that $(x(t)) = ΛAf (x(t)) + ΛI is a Lipschitz continuous function, thus we can give the following assumption.
Assumption 3. There exists a continuous function $ ¯ i (x(t)) such that |$i (x(t))| ≤ d$ ¯ i (x(t)), where d = min {di }.
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1≤i≤n
(30)
Remark 2. Assumption 3 is not restrictive because the upper bound function $ ¯ i (x(t)) is assumed to be unknown. At the same time, $ ¯ i (x(t)) always exists since $(x(t)) is continuous. We can employ FLS (23) to approximate $ ¯ i (t) by
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ˆ $ ¯ i (t) = θiT (t)ψi (t),
(31)
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where ψi (t) is a fuzzy basis function, and θi (t) is an adjustable parameter of the ˆ ¯ i (t) − $ ¯ i (t) } as small as possible. Let the parameter FLS, which makes sup{ $ estimation error be
θ˜i (t) = θi (t) − θi∗ ,
(32)
ˆ εi (t) = $ ¯ i (t) − $ ¯ i (t).
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and let the fuzzy approximation error be (33)
As in the literature [7, 33, 47], it is reasonable to assume that sup{|εi (x(t))|} ≤
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ε∗ for some ε∗ > 0. Consequently, we have
ˆ ˆ ˆ $ ¯ i (t) − $i (t) = $ ¯ i (θi (t), x(t)) − $ ¯ i (t) − εi (t) = θ˜iT (t)ψi (t) − εi (t).
Thus the controller can constructed as following: − uri (t)sign(xi (t)) − µ∗− , xi (t) > 0, ui (t) = 0, xi (t) = 0, − u (t)sign(x (t)) + µ∗ , x (t) < 0, ri i i + 8
(34)
(35)
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with uri = b + θiT (t)ψi (x(t)),
(36)
where b > ∗ .
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Remark 3. Owing to the special form of FONNs (see the term −Cx(t) in (24)), the proposed controller (35) is different from that by conventional AFC methods for integer-order nonlinear systems (see [7]). Moreover, one dose not need to know the sign of the system states.
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Remark 4. Although the condition b ∈ (ε∗ , +∞) is needed in the controller design, it is actually difficult to know the exact value of ε∗ . On the other hand, we may see in section 4 that good control performance can be obtained even if b is small (b = 1), which implies that the proposed control method has good robustness. The FOAL is given as C α 0 Dt θi (t)
= γi |xi (t)|ψi (x(t)), θi (0) ≥ 0,
(37)
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where γi > 0 is the design parameter.
Remark 5. FOAL (37) can also be expressed as:
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1 θi (t) = θi (0) + Γ(α)
Z
0
t
$i (t − ξ)α−1 |xi (ξ)|ψi (x(ξ))dξ.
(38)
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Remark 6. As the term γi |xi (t)|ψi (x(t)) in (37) is not negative, θ(t) ≥ 0 by Lemma 3. As a result, we have uri ≥ 0.
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Theorem 1. Consider the FONN (25) under Assumptions 1-3. The AFC (35) together with the FOAL (37) ensures that lim xi (t) = 0. t→∞
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Proof. Multiplying xT (t) to both sides of (29) we have α T T T xT (t)ΛC 0 Dt x(t) = x (t)$(t) − x (t)ΛCx(t) + x (t)ϕ(u(t))
= −xT (t)ΛCx(t) + ≤ −xT (t)ΛCx(t) +
n X
k=1 n X
k=1
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xk (t)$k (t) + xT (t)ϕ(u(t)) ¯ k (t) + xT (t)ϕ(u(t)). |xk (t)|d$
(39)
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By (34), we obtain n X
k=1
|xk (t)|d$ ¯ k (x(t)) + xT (t)ϕ(u(t)) − xT (t)ΛCx(t)
= xT (t)ϕ(u(t)) − xT (t)ΛCx(t) + d −d
n X
k=1
|xk (t)|θ˜kT (t)ψk (t) + d
n X
k=1
n X
k=1
|xk (t)|εk (t)
≤ −xT (t)ΛCx(t) + xT (t)ϕ(u(t)) + d
k=1
|xk (t)| − d
n X
n X
k=1
|xk (t)|θkT (t)ψk (t)
|xk (t)|θ˜kT (t)ψk (t).
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+ dε∗
n X
|xk (t)|θkT (t)ψk (x(t))
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α xT (t)ΛC 0 Dt x(t) ≤
k=1
(40)
From (35) we know ui ≤ −µ∗− for xi (t) > 0 and ui > µ∗+ for xi (t) ≤ 0. By
(27), we have
(ui (t) + µ∗− )ϕi (ui (t)) ≥ d∗i− u2ri (t) ≥ du2ri (t)
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for xi (t) > 0, and
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(ui (t) − µ∗+ )ϕi (ui (t)) ≥ d∗i+ u2ri (t) ≥ du2ri (t)
(41)
(42)
for xi (t) ≤ 0. Consequently,
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− uri (t)x2i (t)sign(xi (t))ϕi (ui (t)) ≥ du2ri (t)x2i (t) = du2ri (t)|xi (t)|2 .
(43)
xi (t)ϕi (ui (t)) ≤ −duri (t)|xi (t)|.
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That means, for all xi (t),
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(44)
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Substituting (36) and (44) into (40) gives α T xT (t)ΛC 0 Dt x(t) ≤ −duri (t)|xi (t)| − x (t)ΛCx(t) − d
k=1
|xk (t)| + d
= −xT (t)ΛCx(t) − d + d(ε∗ − b)
n X
k=1
n X
k=1
n X
k=1
|xk (t)|θkT (t)ψi (t)
|xk (t)|θ˜kT (t)ψi (t)
|xk (t)|.
(45)
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If we set the design parameter b > ε∗ , then
α T xT (t)ΛC 0 Dt x(t) ≤ −x (t)ΛCx(t) − d
Define
k=1
|xk (t)|θ˜kT (t)ψk (t)
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+ dε∗
n X
n X
n X
k=1
|xk (t)|θ˜kT (t)ψk (t).
(46)
n
ν(t) =
1 T 1 X 1 ˜T ˜ x (t)Λx(t) + θ (t)θk (t). 2d 2 γk k
(47)
k=1
≤
n X 1 ˜T C α ˜ 1 T α x (t)ΛC D x(t) + θ (t)0 Dt θi (t). 0 t d γ i i=1 i
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C α 0 Dt ν(t)
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According to Lemma 2, it holds that
(48)
From (32) we have
α =C 0 Dt θi (t).
(49)
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C α˜ 0 Dt θi (t)
Substituting (37), (46) and (49) into (48), after some straightforward ma-
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nipulators, we have
α 0 Dt V
1 (t) ≤ − xT (t)ΛCx(t). d
(50)
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From (50) and Lemma 5, we know that both xi (t) and θ˜i (t) are bounded. Consequently, θi (t) is bounded. Because uri is bounded by (36), ui (t) is bounded.
Note that ΛC > 0, xi (t) converges to 0 asymptotically (i.e., lim xi (t) = 0). t→∞
Remark 7. (1) AFC method is also used to control FOSs in [23], where the α-th q T derivative of a squared Lyapunov 21 eT (t)Λe(t) is given by 21 (C 0 Dt e(t)) Λe(t) +
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1 C q 2 e(t)Λ(0 Dt e(t))
(see Equality (33)). However, this is incorrect, the correct
form is
α T T C α = (C 0 Dt e(t)) e(t) + e (t)0 Dt e(t) ∞ X Γ(1 + α) α−k C k +2 D e(t)C e(t). 0 Dt Γ(1 + k)Γ(1 − k + α) 0 t
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C α T 0 Dt 2e (t)e(t)
k=1
(2) Actually, we need not to handle the above complicated infinite series. As shown in Theorem 1, we give a strict proof of the stability of the FONNs based on Lemmas 1 and 2 and the proposed Lemmas 3, 4 and 5.
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Remark 8. In [39], adaptive controller has been proposed for FOSs with input nonlinearities. The comparisons between the results of this paper and that of [39] is included in Table 1.
4. Simulation results
In this section three simulation examples will be given to demonstrate the
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effectiveness of the proposed method. 4.1. Example 1
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In system (24), let n = 3, α = 0.97, fi (xi (t)) = tanh(xi (t)), ci = 1, and
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Ii = 0 (i = 1, 2, 3). 2.00 −1.20 0 0.5 0 0 A= 1.71 1.15 2.00 , G = 0 0.5 0 . −4.75 0 1.10 0 0 0.7 The input nonlinearities ϕi (ui (t)) (i = 1, 3) are given as: (ui (t) − 2)(1 − 0.3 sin(ui (t))), ui (t) > 2.0, ϕi (ui (t)) = 0, − 2.0 ≤ ui (t) ≤ 2.0, (u (t) + 2)(0.8 − 0.3 cos(u (t))), u (t) < −2.0. i
i
i
The input nonlinearity ϕ2 (u2 (t)) is chosen as ui (t) > 5, (1 − 0.3 sin(u2 (t)))(ui (t) − 5), ϕ2 (u2 (t)) = 0, − 5 ≤ ui (t) ≤ 5, (0.8 − 0.3 cos(u (t)))(u (t) + 5), u (t) < −5. i i i 12
(51)
(52)
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Table 1: Our results and the results in [39].
The results in [39]
Our results
System model
MIMO FOSs, and prior knowledge
MIMO FONNs, and the structure
should be known
can be fully unknown
Control method
Switching adaptive controller
AFC
Gain reduction toler-
Their
be
Their exact values are not needed
ances
known in advance
Design parameters
There are 3n control parameters to
There are n + 1 control parameters
be chosen
to be chosen
Parameters to be up-
There are 3n parameters that
There are nN parameters that
dated
should be updated online
should be updated
The advantages of the control
The advantages of our method: (1)
methods in [39]: (1) fewer param-
better robustness; (2) the exact
eters to be updated, less computa-
values of the gain reduction toler-
tion burden; (2) the controller is
ances can be unknown (it is dif-
easier to design and may be more
ficult to obtain their exact values
convenient for practical use
[7])
should
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values
AC
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Conclusions
exact
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Comparison
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Under these parameters and without the control input (i.e., ui (t) ≡ 0), the
dynamical behavior of system (24) is shown in Figure 2.
3
4
2 x2(t)
x3(t)
2 0
x2(t)
0
−3
−2
−1 x1(t)
(c)
6
−1
1
0
−2 −4
−3
−2
(d)
6
0
1
4
x3(t)
4
−1 x1(t)
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2
x3(t)
1 0
4
2 0
2 0
−3
−2
−1 x1(t)
0
−2 −2
1
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−2 −4
(b)
4
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(a)
0
2
4
x2(t)
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Figure 2: Dynamical behavior of system (24) in (a) x1 (t) − x2 (t) − x3 (t), (b) x1 (t) − x2 (t) plane, (c) x1 (t) − x3 (t) plane and (d) x1 (t) − x4 (t) plane.
It is worth mentioning, within all simulation studies, the proposed con-
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trol method does not need the knowledge of the FONNs’s model. Indeed, the FONNs’s model is only given for simulation purposes. The FLSs use x1 (t), x2 (t)
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and x3 (t) as their inputs. From Figure 2 we know that the system variables distribute on interval [-3, 3]. Thus, for each input variable, we define 3 Gaussian membership functions distribute uniformly on interval [-3, 3]. As a result, there are 3 × 3 × 3 = 27 fuzzy rules are used in the simulation. The Gaussian mem-
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bership functions are given in Figure 3. The design parameters are chosen as b = 1, γ1 = γ2 = γ3 = 10, x(0) = [2.0, −2.0, 3.0]T , θi (0) = 0 ∈ R27 .
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1
0.6
0.4
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Membership functions
0.8
0.2
0 −3
−2
−1
0 xi(t),i=1,2,3
1
2
3
Figure 3: Membership functions.
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In the simulation, sign(·) is replaced by arctan(10·). Our results are given in Figures 4, 5, and 6. These results indicate that good control performance has been achieved. Figure 4 gives the state variables. Figure 5 presents the control inputs. Noting that the input nonlinearities are given as (51) and (52), the control input u2 (t) tends to 5, and ui (t), i = 1, 3 tend to -2 eventually. Figure 6 depicts the input nonlinearities. It should be mentioned that, although b is set very small, the states converge to 0 rapidly. That is to say that the employed
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FLSs have good approximation abilities.
x1(t)
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3
x2(t) x3(t)
1
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State variables
2
0
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−1
−2
0
1
2
3
4
5 6 Time(second)
7
Figure 4: The state variables.
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10 u1(t) u2(t) u3(t)
0
−5
−10
0
1
2
3
4
5 6 Time(second)
7
8
6
Φ2(u2(t)) Φ3(u3(t))
2 0 −2
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Input nonlinearities
10
Φ1(u1(t))
4
−4 −6
0
1
2
3
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−8
9
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Figure 5: Control inputs.
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Control inputs
5
4
5 6 Time(second)
7
8
9
10
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Figure 6: The input nonlinearities.
In the remainder of this subsection, we will show how different choices of membership functions and fuzzy rules affect the control performance. We give
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four examples presented in Table 2 (the step is 0.02) by using different membership functions and fuzzy rules. The simulation results are included in Figure 7. In this paper, the fuzzy membership functions are chosen according to expertise
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and control performance. It should be mentioned that similar functions are used in [6, 7, 21, 22, 23, 26, 33, 45, 46]. According to simulation results in Figure 7, the proposed AFC works well even when different membership functions are used. It can be concluded that, in all these cases, the simulation results are similar. However, when more fuzzy membership functions are chosen, more fuzzy rules and simulation time are needed. Moreover, the triangular membership function
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computation will consume more time than the Gaussian function. Thus, in the beginning of this subsection, we use three Gaussian functions in the simulation. Further investigation is needed to determine the optimal number of membership
Table 2: Different membership functions
Member functions
Parameters
Fuzzy rules
Time consumed
3 Gaussian functions
[1.2, i], i = 1, 2, 3
27
1.52 seconds
5 Gaussian functions
[0.4, i], i = 1, · · · , 5
125
5 Triangular functions
[i − 2, i, i + 2], i = 1, · · · , 5
125
[0.2, i], i = 1, · · · , 7
343
Case 1: 3 Gaussian functions
4
0
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2
x1(t) x3(t)
0
0
x3(t)
0 −1 0
5
10
Case 4: 7 Gaussian functions x1(t)
3
x2(t)
−1 −2
x2(t)
1
4
1
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State variables
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3
x1(t)
Time(second)
Case 3: 5 triangular functions
4
2
−2
10
State variables
−2
State variables
x3(t)
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State variables
2
−1
51.63 seconds
3
x2(t)
0
10.63 seconds
Case 2: 5 Gaussian functions
4
x1(t)
3
1
7.88 seconds
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7 Gaussian functions
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functions and their parameters to achieve better control performance.
x2(t)
2
x3(t)
1 0 −1
5 Time(second)
−2
10
0
5
10
Time(second)
Figure 7: Simulation results for different choices of membership functions.
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4.2. Example 2 Suppose that the FONNs have time-varying gain aij (t) (i.e., the matrix A is replaced by A(t)) and external disturbances [5], which is more complicated
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than above model: α = 0.98, fi (xi (t)) = tanh(|x i (t)| − 1), ci = 1, Ii = 0, a11 (x1 (t)) 1 −9 1.99, |xi (t)| ≤ 1, , aii (xi (t)) = A= −9 a22 (x2 (t)) 1 −1.99, |x (t)| > 1. i 1 −9 a33 (x3 (t)) To show the robustness of the control system, let us add three external distur-
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bances to system (24): d1 (t) = 0.48 sin t, d2 (t) = −0.48 cos t, d3 (t) = 0.49 sin t − 1.2 −0.2 0.1 0.5 cos t. The control gain matrix is chosen as G = −0.2 1.2 0.1 , which 0.1 0.1 1.7 is a positive definite matrix. The FLSs has control parameters and input nonlinearities chosen as above. Let x(0) = [−3, 2, −3] and θi (0) = 0 ∈ R27 , i = 1, 2, 3. Our results are presented in Figures 8 and 9. From Figure 8 we can see that
the state variables converge to the origin very fast, and have small fluctuations. Figure 9 depicts the time response of the input nonlinearities. The qualitative
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analysis of these results is the same as that of Example 1.
1 0
x2(t) x3(t)
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State variables
x1(t)
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2
−1
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−2 −3
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−4
0
1
2
3
4
5 6 Time(second)
Figure 8: The state variables.
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8
9
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5 Φ1(u1(t))
0
−5
0
1
2
3
4
5 6 Time(second)
7
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Φ3(u3(t))
8
9
10
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Input nonlinearities
Φ2(u2(t))
Figure 9: The input nonlinearities.
4.3. Example 3
It should be mentioned that our controller is also valid for many classes of
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FOSs. We will give a comparison between our control method and that of [39]. The mathematical model of the uncertain FOS used in [39] (see Equality (76)
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on page 218 in [39]) can be described as C α 0 Dt x1 (t) =10(x2 (t) − x1 (t)) + x4 (t) + 0.2 cos(3x1 (t))x1 (t) + ϕj (u1 (t)), α C 0 Dt x2 (t) =28x1 (t) − x2 (t) + x1 (t)x3 (t) + 0.12 sin(4x2 (t))x2 (t) + ϕ2 (u2 (t)), (53)
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8 C α 0 Dt x3 (t) =x1 (t)x2 (t) − x3 (t) + 0.12 sin(4x2 (t))x2 (t) + ϕ3 (u3 (t)), 3 C Dα x (t) = − x (t)x (t) − x (t) + 0.1 cos(2x (t))x (t) + ϕ (u (t)), 2 3 4 4 4 4 4 0 t 4
with α = 0.93.
Let the initial condition be x(0) = [2, 4, 1, 3]T . The input nonlinearities
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ϕi (ui (t)) (i = 1, 3) are given as (51). The control parameters in [39] are chosen as ci = 0.25, ηi = 8.24, κi λi = 6 (see Equalities (78) and (79) in [39]). Our control parameters are chosen the same as that in Example 2. The simulation results are presented in Figures 10 and 11. The simulation results indicate that under the chosen parameters, compared with the control method in [39], our method has smaller control inputs and faster convergence of the state vector.
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In (53), d(t) = [0.2x1 cos 3x1 , 0.12x2 sin 4x2 , 0.12x2 sin 4x2 , 0.1x4 cos 2x4 ]T . To show the robustness of the proposed control method, let us use larger disturbance: cd(t) where c is a positive constant. The simulation results are shown in Figure 12. It can be concluded that by using the control method in [39],
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when the disturbance is chosen as 3d(t), the system variables needs more time
to converge to the origin. When the disturbance is chosen as 6d(t), the controller proposed by [39] will be not valid anymore (to overcome this problem, one can choose larger control parameters). However, our control method works
well when the disturbance changes largely. That is to say, the proposed AFC
(a)
x 2 (t) x 3 (t) x 4 (t)
2 0
0.5
1
1.5
Time(second) (c)
6
x 1 (t)
4 2
0
0.5
1
1.5
x 3 (t) x 4 (t)
2 1
0
Time(second)
x 2 (t) x 3 (t) x 4 (t)
0.4
0.6
x 1 (t)
4
x 2 (t) x 3 (t)
2
x 4 (t)
0 -2
2
0.2
x 1 (t)
Time(second) (d)
6
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0
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x 2 (t)
(b)
×10 284
3
0
2
State variables
4
State variables
x 1 (t)
6
0
State variables
4
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State variables
8
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method has satisfactory robustness.
0
0.5
1
1.5
2
Time(second)
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Figure 12: Our results and the results in [39] with larger disturbance: (a) the results of [39] when c = 3; (b) the results of [39] when c = 6; (c) our results when c = 3; (d) our results
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when c = 6.
5. Conclusions An AFC for a class of FONNs with input nonlinearities and dead-zones has
been proposed in the paper, where the unknown structure of the FONNs and the exact values of gain reduction tolerances are approximated by FLSs. By using the proposed FOAL, the stability of the closed-loop system is proven strictly 20
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based on fractional Lyapunov criteria. By using the results of this study, many control methods that are valid for integer-order NNs can be extended to control FONNs. Simulation studies have demonstrated that the proposed method is not only efficient for obtaining parameter convergence but also useful for improving
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the control performance and robustness. Future work will be focused on how
to choose appropriate fuzzy rules and membership functions to achieve better control performance. 6. Acknowledgements
This work is supported by the National Natural Science Foundation of China
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under Grant 11771263, and the Natural Science Foundation of Anhui Province
of China under Grant 1808085MF181. The authors would like to thank the editor and the anonymous referees for their comments and suggestions. References
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Figure 1: Input nonlinearity.
(a)
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(b)
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x 2 (t)
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x 1 (t)
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Time(second) (c)
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0
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Figure 10: The time response of the state variables under our control approach (solid line)
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and the method of [39] (dashed line) in (a) x1 (t); (b) x2 (t); (c) x3 (t); (d) x4 (t).
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(a)
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(b)
100
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u 1 (t)
50 0
0
-50
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0.5
1
Time(second) (c)
0 -20 -40
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u 3 (t)
-100
2
0.5
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1.5
0
0
1
2
Time(second)
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Time(second)
2
0
-50
2
1
Time(second) (d)
50
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20
-60
1.5
u 4 (t)
-50
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Figure 11: The time response of the control inputs under our control approach (solid line)
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and the method of [39] (dashed line) in (a) u1 (t); (b) u2 (t); (c) u3 (t); (d) u4 (t).
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